1896] GENERALIZATION OF FERMAT'S THEOREM.

189

A TWO-FOLD GENERALIZATION OF FERMAT'S THEOREM.

Presented to the American Mathematical Society, February 29, 1896.

BY PROFESSOR ELIAKIM HASTINGS MOORE.

Formulation of the generalized Fermat theorem III [k+1, n; p]. SS 1-4.

I,

1. In Gauss's congruence notation Fermat's theorem is: aP-a = 0 (mod p)

where p is any prime and a is any integer:

or, otherwise expressed,

I The two rational integral functions of the indeterminate X with integral coefficients

I The two forms in the two indeterminates X, X1,

3

D[2, 1; p] (X, X1) = X X,”— XX1,

a-p-1

P[2, 1; p] (X, X1)=X. II (aX+X,),

are identically congruent (mod p):

D[2, 1 ; p] (X, X1) =P[2, 1 ; p] (X, X1) (mod p ).

2. We proceed in two steps to a two-fold generalization

of Fermat's theorem I.

...

II. The two forms in the k+1 indeterminates X, X1,···, Xx (1) D[k+1,1; p] (X, X,,···, X, ̧)= | X3' |

X1,

(2) P[k+1,1;p] ( X, X1, ···, X1)= [I* £a,X,

(i, j = 0, 1,...,k), (g = 0, 1,...,k),

—where the product П* embraces the (p+1-1)/(p—1) linear

g=k

g=0

forms a, X, whose coefficients a, (g=0, 1, ..., k) are integers selected from the series 0, 1,, p-1, in all possible ways, only

so that for every particular form, first, the coefficients a are not all 0, and, second, of the coefficients a not 0 the one with largest index g is 1 are identically congruent (mod p):

D[k+1, n ; p] (X, X,1, ···, X)= P [k+1, n; p] ( X, X,···, X.).

19

...

When we collect into a class the totality of integers congruent to one another (mod p), and denote the p incongruent classes by p marks, we have in this system of p marks a field F[p] of order p and rank 1. The marks of the field F[p] may be combined by the four fundamental operations of algebra-addition, subtraction, multiplication, division, -the operations being subject to the ordinary abstract operational laws of algebra, the results of these operations being in every case uniquely determined and belonging to the field. Congruences (=) (mod p) are in the field equalities (=), and identical congruencies (E) are identities (=). The restatement of II in the terminology of the field F[p] is given by setting n=1 in its generalization III (§3).

3. The second step of generalization of I, rests upon. Galois's generalization of the field F[p] to the Galois-field GF [p"] of order p", modulus p, and rank n. This field of p" marks is uniquely defined for every p=prime, n=positive integer. (I have elsewhere proved that every field of finite orders is a Galois-field of order s=p".)

a

A form, that is, a rational integral function of certain indeterminates, .,,,, is said to belong to the GF [p"] if its coefficients belong to (are marks a of) the GF[p"]. A

g=k

linear homogeneous form Σ 4X, belonging to the GF[p"] is

g=0

g

called primitive if not all its coefficients a are 9, and if of the coefficients a not 0 the one with largest index g is 1. We have then":

9

...

III. The two forms in the k+1 indeterminates, X, X1, ···, X ̧, (3) D[k+1,n; p] (X, X,, I)= I

...

(4) P[k+1,n; p] (X, X,,···, X)= II* Σ«, X,

g

(i, j=0, 1,..., k), (g=0, 1,, k),

—where the product II* embraces the (p"(k+1)—1) / (p”—1) dis

are identical:

g=k

99

tinct primitive linear homogeneous forms a belonging to the GF[p"] D[k+1, n; p] (X, X, ̈‚ X1)= P[k+1, n; p] (X, X1, ···, Ã ̧ ).

...

The forms D, P whose identity theorem III affirms have the three characteristic positive integers or characters k + 1, n, p. It is convenient to attach these characters to the no